Linear and multilinear algebra; matrix theory Definition

Moore–Penrose inverse

Definition

Let $A$ be an $m \times n$ complex matrix. There is exactly one $n \times m$ complex matrix $A^+$ with these four properties:

$A A^+ A = A$ .
$A^+ A A^+ = A^+$ .
$(A A^+)^* = A A^+$ .
$(A^+ A)^* = A^+ A$ .

This matrix is the Moore–Penrose inverse, or pseudoinverse, of $A$ , where $^*$ is the conjugate transpose.

Remarks

If $A$ is invertible, then $A^+ = A^{-1}$ . If the columns of $A$ are linearly independent, then $A^+ = (A^* A)^{-1} A^*$ . For $b \in \C^m$ , the vector $A^+ b$ is the least squares solution of $Ax = b$ with the least norm. If $A = U \Sigma V^*$ is a singular value decomposition, then $A^+ = V \Sigma^+ U^*$ , where $\Sigma^+$ inverts the nonzero singular values and transposes.

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Matrix definition
Transpose definition
Conjugate transpose definition
Moore–Penrose inverse definition

Tag	`0PSRCMCM5X`
URI	`m2://0PSRCMCM5X`
Kind	definition
Writing	W1 stated
MSC	15A09
Defines	Moore–Penrose inverse
Uses	Conjugate transpose

Moore–Penrose inverse

Definition

Remarks

Reading path

Further reading

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